Ode to Prime Numbers

“No branch of number theory is more saturated with mystery and elegance than the study of prime numbers,” wrote Martin Gardner in his essay “Patterns and Primes.” It is therefore no wonder that prime numbers show up in another human endeavor that delves into mysteries in search of patterns and elegance—poetry. As a mathematician and poet, I have long been interested in this confluence.

Some poems, echoing the purpose of early poetic treatises on scientific principles, attempt to elucidate the mathematical concepts that underlie prime numbers. Others play with primes’ cultural associations. Still others derive their structure from mathematical patterns involving primes. Whatever the mode of introduction, the meeting of poetry and primes—“those exasperating, unruly integers that refuse to be divided evenly by any integer except themselves and 1,” as Gardner described them—is often an eventful one.

Poetic Mathematics

Gardner often quoted poems in his Mathematical Games column for Scientific American, and he wrote several essays on prime numbers. He could hardly have found a better poem for the subject than British poet Helen Spalding’s “Let Us Now Praise Prime Numbers,” which he reprinted in the essay “Strong Laws of Small Primes.” The poem captures elements that have made primes an object of fascination since the time of Euclid. Spalding (1920–1991) is herself a mysterious figure whose life is difficult to trace after her last publication in The London Magazine in 1961.

Let Us Now Praise Prime Numbers

Let us now praise prime numbers
With our fathers who begat us:
The power, the peculiar glory of prime numbers
Is that nothing begat them,
No ancestors, no factors,
Adams among the multiplied generations.

None can foretell their coming.
Among the ordinal numbers
They do not reserve their seats, arrive unexpected.
Along the lines of cardinals
They rise like surprising pontiffs,
Each absolute, inscrutable, self-elected.

In the beginning where chaos
Ends and zero resolves,
They crowd the foreground prodigal as forest,
But middle distance thins them,
Far distance to infinity
Yields them rare as unreturning comets.

The poem’s first stanza alludes to the Fundamental Theorem of Arithmetic. This theorem states that every positive integer greater than 1 is either a prime number or can be expressed as a unique product of prime numbers. Thus the primes are the building blocks of the integers and, consequently, of the entire real number system. In the second and third stanzas, Spalding suggests how prime numbers appear among the other numbers: Scattered without a discernible pattern, they fan out and occur less frequently as the numbers grow larger. However, despite this reduction in frequency, an infinite number of primes exists. Euclid’s proof of the infinitude of prime numbers, circa 300 BCE, is considered to be one of the most elegant proofs in mathematics—a poem in its own right. Michael Szpakowski’s Proof, a Short Opera offers a poetic and musical rendition of this proof. The piece can be viewed at www.somedancersandmusicians.com/proof/.

In the poem’s final stanza, Spalding touches on one of the deep mysteries associated with prime numbers: our inability to pin them down with a formula. Prime numbers smaller than a given number N can be found through a technique called the Sieve of Eratosthenes—named for Eratosthenes (ca. 276–195 BCE), the Greek mathematician who discovered it. The “sifting” consists of a simple divisibility test and the systematic deletion of all the proper multiples of the prime numbers up to the largest prime smaller than the square root of N. The method works best when N itself is small. For N = 100, for example, the deletion leaves in the sieve the first 25 primes:

Since the time of Eratosthenes, many techniques have been invented to “catch” prime numbers, but as yet no formula has been found that covers them all. In particular, it is notoriously difficult to produce very large primes. Neither has a pattern been found to predict their distribution within a given interval of numbers. In 2000, the Clay Mathematics Institute listed seven of the most important open problems in mathematics. The institute offers an award of $1 million to anyone who publishes a solution to one of these Millennium Prize Problems. One problem, the Riemann Hypothesis, formulated by Bernhard Riemann (1826–1866), celebrated its 150th anniversary in 2010. It is a conjecture about the zeros of the Riemann zeta function. The function, ζ, is defined for complex variables, s, and a value of s for which ζ (s) = 0 is called a zero of zeta. The zeta function was introduced by Leonhard Euler in the early 1800s as a function of a real variable. Riemann extended the function to complex numbers and established a connection between its set of zeros and properties of prime numbers. The Riemann Hypothesis is considered to be the most important open problem in pure mathematics, and its solution would advance our knowledge of the distribution of prime numbers. Tom Apostol’s poem, “Where Are the Zeros of Zeta of s?,” playfully imparts the excitement generated by the chase after its solution. It begins:

Where are the zeros of zeta of s?G. F. B. Riemann has made a good guess;
They’re all on the critical line, saith he,
And their density’s one over 2pi log t.

This statement of Riemann’s has been like a trigger
And many good men, with vim and with vigor,
Have attempted to find, with mathematical rigor,
What happens to zeta as mod t gets bigger.

—Tom Apostol, from “Where Are the Zeros of Zeta of s?”

Many other questions about prime numbers remain unanswered. Some of these problems and their partial solutions, as well as the spell cast by primes on the mathematicians who study them, have also made their way into poetry.